Question Find the derivative of the function $h(x) = e^{2x^2} - 9x + 3/x$ .

Studdy Solution

STEP 1

Assumptions
1. The function given is $h(x) = e^{2x^2} - 9x + \frac{3}{x}$ .
2. We need to find the derivative of $h(x)$ with respect to $x$ , denoted as $h'(x)$ .
3. We will use the rules of differentiation, including the power rule, the exponential rule, the product rule, and the quotient rule.

STEP 2

To find $h'(x)$ , we will differentiate each term of $h(x)$ separately. The function $h(x)$ consists of three terms: $e^{2x^2}$ , $-9x$ , and $\frac{3}{x}$ .

STEP 3

Differentiate the first term, $e^{2x^2}$ , using the chain rule. The chain rule states that the derivative of $e^{u(x)}$ is $e^{u(x)} \cdot u'(x)$ , where $u(x)$ is a function of $x$ .

STEP 4

Let $u(x) = 2x^2$ . Then, $u'(x)$ is the derivative of $u(x)$ with respect to $x$ .

STEP 5

Apply the power rule to find $u'(x)$ . The power rule states that the derivative of $x^n$ is $nx^{n-1}$ .

STEP 6

Calculate $u'(x)$ .
$u'(x) = \frac{d}{dx}(2x^2) = 2 \cdot 2x^{2-1} = 4x$

STEP 7

Now apply the chain rule to find the derivative of $e^{2x^2}$ .
$\frac{d}{dx}(e^{2x^2}) = e^{2x^2} \cdot 4x$

STEP 8

Differentiate the second term, $-9x$ , using the power rule.
$\frac{d}{dx}(-9x) = -9$

STEP 9

Differentiate the third term, $\frac{3}{x}$ , using the quotient rule or recognizing it as $3x^{-1}$ and using the power rule.

STEP 10

If using the power rule, rewrite $\frac{3}{x}$ as $3x^{-1}$ .

STEP 11

Apply the power rule to find the derivative of $3x^{-1}$ .
$\frac{d}{dx}(3x^{-1}) = 3 \cdot (-1)x^{-1-1} = -3x^{-2}$

STEP 12

Now, combine the derivatives of all three terms to find $h'(x)$ .
$h'(x) = e^{2x^2} \cdot 4x - 9 - \frac{3}{x^2}$

STEP 13

Simplify the expression for $h'(x)$ if necessary.
$h'(x) = 4xe^{2x^2} - 9 - \frac{3}{x^2}$
This is the derivative of the function $h(x)$ .

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